PATH-INTEGRALS FOR SPINNING PARTICLES, STATIONARY-PHASE AND THE DUISTERMAAT-HECKMANN THEOREM

We examine the problem of the evaluation of both the propagator and of the partition function of a spinning particle in an external held at the classical as well as the quantum level, in connection with the ass erted exactness of the stationary phase approximation. At the classica l level we argue that exactness of this approximation stems from the f act that the dynamics (on the two-sphere S-2) of a spinning particle i n a magnetic field is the reduction from R(4) to S-2 of a linear dynam ical system on R(4). At the quantum level, however, and within the pat h integral approach, the restriction, inherent to the use of the stati onary phase approximation, to regular paths clashes with the fact that no regulators are present in the action that enters the path integral . This is shown to lead to a prefactor for the path integral that is s trictly divergent, except in the classical limit. A critical compariso n is made with the various approaches that have been presented in the literature. The validity of a formula given in literature for the spin propagator is extended to the case of motion in an arbitrary magnetic field. (C) 1996 American Institute of Physics.